Problem: Hiroki and Mapiya were asked to find an explicit formula for the sequence $125\,,\,25\,,\,5\,,\,1,...$, where the first term should be $f(1)$. Hiroki said the formula is $f(n)=625\cdot\left(\dfrac{1}{5}\right)^{{n}}$, and Mapiya said the formula is $f(n)=125\cdot\left(\dfrac{1}{5}\right)^{{n}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Hiroki (Choice B) B Only Mapiya (Choice C) C Both Hiroki and Mapiya (Choice D) D Neither Hiroki nor Mapiya
Solution: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{1}{5}=\dfrac{5}{25}=\dfrac{25}{125}={\dfrac{1}{5}}$ We see that the constant ratio between successive terms is ${\dfrac{1}{5}}$. In other words, we can find any term by starting with the first term and multiplying by ${\dfrac{1}{5}}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $f(n)$ ${125}\cdot\!\left({\dfrac{1}{5}}\right)^{0}$ ${125}\cdot\!\left({\dfrac{1}{5}}\right)^{1}$ ${125}\cdot\!\left({\dfrac{1}{5}}\right)^{2}$ ${125}\cdot\!\left({\dfrac{1}{5}}\right)^{3}$ We can see that every term is the product of the first term, ${125}$, and a power of the constant ratio, ${\dfrac{1}{5}}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${125}$ is the first term and ${\dfrac{1}{5}}$ is the constant ratio): $f(n)={125}\cdot\left({\dfrac{1}{5}}\right)^{{\,n-1}}$ We can now expand this formula: $ \begin{aligned} f(n)= &{125}\cdot\left({\dfrac{1}{5}}\right)^{{\,n-1}}\\\\ = & 125\cdot\left(\dfrac{1}{5}\right)^{{\,n}}\cdot \left(\dfrac{1}{5}\right)^{-1}\\\\ = & 125\cdot 5\cdot\left(\dfrac{1}{5}\right)^{{\,n}}\\\\ = &625\cdot \left(\dfrac{1}{5}\right)^{{n}}\end{aligned}$ So Hiroki is definitely right. What about Mapiya? We can see that $f(n)=125\cdot \left(\dfrac{1}{5}\right)^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $f(1)=125\cdot \left(\dfrac{1}{5}\right)^{{\,1}} = 25$. However, according to our table of values, $f(1)=125$. So Mapiya is definitely wrong. Only Hiroki got a correct explicit formula.